Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization...

Review

• Taylor Series and Error Analysis• Roots of Equations• Linear Algebraic Equations• Optimization• Numerical Differentiation and Integration• Ordinary Differential Equations• Partial Differential Equations• Curve Fitting

Numerical MethodsLecture 22

Prof. Jinbo BiCSE, UConn

Taylor Series

• Lagrange remainder

Roots of Equations

• Bracketing Methods• Bisection Method

• False Position Method

• Open Methods• Fixed Point Iteration

• Newton-Raphson Method

• Secant Method

• Roots of Polynomials• Müller’s Method

• Bairstow’s Method

Bisection Method

• Example:

• Use range of [202:204]

• Root is in upper subinterval

Bisection Method

• Use range of [203:204]

• Root is in lower subintervalNumerical MethodsLecture 22

Fixed Point Iteration Example

Special attention

Read Chap 6.1, 6.6

Newton-Raphson Method

• Use tangent to guide you to the root

Linear Algebraic Systems

• Gaussian Elimination• Forward Elimination• Back Substitution

• LU Decomposition

Gaussian Elimination

• Forward elimination

• Eliminate x1 from row 2

• Multiply row 1 by a21/a11

• Eliminate x1 from row 2• Subtract row 1 from row 2

• Eliminate x1 from all other rows in the same way

• Then eliminate x2 from rows 3-n and so onNumerical MethodsLecture 22

• Forward elimination

• Back substitute to solve for x

LU Decomposition

• Substitute the factorization into the linear system

• We have transformed the problem into two steps• Factorize A into L and U• Solve the two sub-problems

• LD = B• UX = D

LU Decomposition

• Example

• Factorize A using forward elimination

LU Decomposition

• Example

LU Decomposition

• Example

LU Decomposition

• Example

Optimization Methods• One-dimensional unconstrained optimization

• Golden-Section

• Quadratic Interpolation

• Newton’s Method

• Multidimensional unconstrained optimization• Direct Methods

• Gradient Methods

• Constrained Optimization• Linear Programming

Golden-section search

• Algorithm• Pick two interior points in the interval using the

golden ratio

• Two possibilities

• Example

Newton’s Method

• Newton-Raphson could be used to find the root of an function

• When finding a function optimum, use the fact that we want to find the root of the derivative and apply Newton-Raphson

Newton’s Method

• Example

Newton’s Method

• Example

Quadratic interpolation

• Use a second order polynomial as an approximation of the function near the optimum

Special attention

Gradient Methods

• Given a starting point, use the gradient to tell you which direction to proceed

• The gradient gives you the largest slope out from the current position

Special attention

Numerical Integration• Newton-Cotes

• Trapezoidal Rule• Simpson’s Rules (Special attention for

unevenly distributed points)

• Romberg Integration• Gauss Quadrature

Newton-Cotes Formulas• Trapezoidal Rule

• Simpson’s 1/3 Rule

• Simpson’s 3/8 Rule

Special attention

Read Chap 21.2-3

Integration of Equations

• Romberg Integration• Use two estimates of integration and then

extrapolate to get a better estimate

• Special case where you always halve the interval - i.e. h2=h1/2

Romberg Integration

Ordinary Differential Equations• Runge-Kutta Methods

• Euler’s Method• Heun’s Method• RK4

• Multistep Methods• Boundary Value Problems• Eigenvalue Problems

Euler’s Method

• Example:

• True:• h=0.5

Heun’s Method

• Local truncation error is O(h3) and global truncation error is O(h2)

Heun’s Method

Classic 4th-order R-K method

Special attention to

ODE equation system

Not only one equation

Curve Fitting

• Least Squares Regression• Interpolation• Fourier Approximation

Polynomial Regression• An mth order polynomial will require that you

solve a system of m+1 linear equations

• Standard error

Special attention

Lecture note 19

Chap 17.1

Newton (divided difference) Interpolation polynomials

Interpolation• General Scheme for Divided Difference

Coefficients

Interpolation

• General Scheme for Divided Difference Coefficients

Interpolation

• Example:• Estimate ln 2 with data points at (1,0),

(4,1.386294)• Linear interpolation

Interpolation

(4,1.386294), (5,1.609438)• Quadratic interpolation

Interpolation

(4,1.386294), (5,1.609438), (6,1.791759)• Cubic interpolation

Spline Interpolation

• Spline interpolation applies low-order polynomial to connect two neighboring points and uses it to interpolate between them.

• Typical Spline functions

Cubic Spline Functions

• This gives us n-1 equation with n-1 unknowns – the second derivatives

• Once we solve for the second derivatives, we can plug it into the Lagrange interpolating polynomial for second derivative to solve for the splines

• Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5)• At x=x1=4

• At x=x2=7

• Solve the system of equations to find the second derivatives

Cubic Spline Equations

• Substituting for other intervals

Final Exam• December 13 Friday, 10:30 AM~12:30

PM, ITE 119• Closed book, three cheat sheets

(8.5x11in) allowed • Office hours:

• December 12, 1-3pm, or by appointment• TA December 10, 11am-12noon or by

appointment

Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization...

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